Volume 372, Issue 24, 9 June 2008, Pages 4344-4349
Roman Tomaschitz^{}^{, }^{a}^{, }^{}
Abstract
Tachyonic spectral densities of ultra-relativistic electron populations are fitted to the γ-ray spectra of two TeV blazars, the BL Lacertae objects 1ES 0229+200 and 1ES 0347-121. The spectral maps are compared to Galactic TeV sources, the γ-ray binary LS 5039 and the supernova remnant W28. In contrast to TeV photons, the extragalactic tachyon flux is not attenuated by interaction with the cosmic background light; there is no absorption of tachyonic γ-rays via pair creation, as tachyons do not interact with infrared background photons. The curvature of the observed γ-ray spectra is intrinsic, caused by the Boltzmann factor of the electron densities, and reproduced by a tachyonic cascade fit. In particular, the curvature in the spectral map of the Galactic microquasar is more pronounced than of the two extragalactic γ-ray sources. Estimates of the thermodynamic parameters of the thermal or, in the case of supernova remnant W28, shock-heated nonthermal electron plasma generating the tachyon flux are obtained from the spectral fits.
Keywords: Superluminal radiation; Tachyonic cascade spectra; Ultra-relativistic electron plasma; TeV blazars; Spectral averaging; Proca equation with negative mass-square
PACS classification codes: 03.50.Kk; 05.30.Fk; 52.27.Ny; 95.30.Gv
Article Outline
1. Introduction
The goal is to point out evidence for superluminal γ-rays from two active galactic nuclei, the BL Lacertae objects 1ES 0229+200, cf. Refs. [1], [2], [3] and [4], and 1ES 0347-121, cf. Refs. [5], [6] and [7]. Spectral maps of these TeV blazars have recently been obtained by means of ground-based imaging air Cherenkov detectors [4] and [7]. Here, a tachyonic cascade fit is performed to the γ-ray spectrum of these blazars. In contrast to electromagnetic γ-rays, tachyons cannot interact with infrared background photons, so that there is no attenuation of the extragalactic tachyon flux by electron–positron pair production. The observed spectrum is already the intrinsic one without any need of absorption correction as required in electromagnetic spectral fits [8] and [9]. The spectral curvature is generated by the Boltzmann factor of the thermal electron plasma in the galactic nuclei.Tachyonic cascade spectra are obtained by averaging the superluminal spectral densities of individual electrons with ultra-relativistic Fermi distributions. We use the averaged radiation densities to perform spectral fits to the γ-ray spectra of the mentioned BL Lacertae objects as well as the microquasar LS 5039 and the supernova remnant W28. Tachyonic cascade spectra generated by thermal electron populations in the active galactic nuclei (AGNs) provide excellent fits to the observed γ-ray flux. The temperature, source count, and internal energy of the ultra-relativistic electron plasma are obtained from the spectral fits. The spectral map of the Galactic microquasar is no less curved than the cascades in the AGN spectra. Moreover, the spectral curvature of the AGNs does not increase with increasing redshift, which is further evidence for an unattenuated extragalactic γ-ray flux.
The tachyonic radiation field is a Proca field with negative
mass-square,
In Section 2, we further elaborate on the tachyonic Proca equation by comparing to electromagnetic theory, and assemble the tachyonic spectral averages employed in the fits, which are based on the transversal and longitudinal radiation densities generated by a free electronic spinor current. In Section 3, the spectral fitting is explained, and intrinsic spectral curvature is argued by comparing the cascade spectra of the above-mentioned Galactic and extragalactic TeV sources. The conclusions are summarized in Section 4.
2. Spectral maps in the γ-ray bands: Origin and structure of tachyonic cascade spectra
The Lagrangian (1) resembles
its electrodynamic counterpart, but the negative mass-square of the
Proca field causes striking differences. Apart from the superluminal
speed of the tachyonic quanta, the radiation is partially
longitudinally polarized [12], the gauge
freedom is broken, and freely propagating charges can radiate
superluminal quanta. The analogy to Maxwell's theory becomes even more
transparent in 3D. The tachyonic E and B
fields are related to the vector potential A_{α}=(A_{0},A)
by E=A_{0}−∂A/∂t
and B=rotA,
and the field equations decompose into
The spectral fits in Section 3 are based on
the quantized tachyonic radiation densities
Only frequencies in the range 0ωω_{max}(γ) can be radiated by a uniformly moving electron, the tachyonic spectral densities p^{T,L}(ω,γ) being cut off at the break frequency ω_{max}. A positive ω_{max}(γ) requires Lorentz factors exceeding the threshold μ_{t} in (4), since ω_{max}(μ_{t})=0. The lower threshold on the speed of the electron for radiation to occur is thus υ_{min}=m_{t}/(2mμ_{t}). The tachyon–electron mass ratio gives υ_{min}/c≈2.1×10^{−3}, which is roughly the speed of the Galaxy in the microwave background [17].
The radiation densities (3) refer to a
single spinning charge with Lorentz factor γ; we
average them with a Fermi power-law distribution [18] and [19],
Fig. 1. Spectral map of the BL Lac 1ES 0229+200. HESS data points from [4]. The solid line T+L depicts the unpolarized differential tachyon flux dN^{T+L}/dE, obtained by adding the flux densities ρ_{1,2} of two electron populations, cf. (16) and Table 1. The transversal (T, dot-dashed) and longitudinal (L, double-dot-dashed) flux densities add up to the total unpolarized flux T+L. The exponential decay of the cascades ρ_{1,2} sets in at about E_{cut}≈(m_{t}/m)kT, implying cutoffs at 3.6 TeV for the ρ_{1} cascade and 120 GeV for ρ_{2}.
Fig. 2. Spectral map of the BL Lac 1ES 0347-121. HESS points from [7]. The unpolarized spectral fit T+L is based on the electron distributions recorded in Table 1, the polarized flux components are labeled T and L. The ρ_{1} cascade is cut at E_{cut}≈4.0 TeV, and ρ_{2} at 190 GeV. The curvature in the spectral slope of 1ES 0347-121 at z≈0.188 is less pronounced than of 1ES 0229+200 at z≈0.140, suggesting that the shape of the rescaled flux density is intrinsic rather than generated by intergalactic absorption.
Fig. 3. Spectral map of the γ-ray binary LS 5039 close to periastron. HESS data points at the superior conjunction [32]. Notation as in Fig. 1. The cascade ρ_{1} is exponentially cut at E_{cut}≈6.3 TeV, and ρ_{2} at 190 GeV, cf. Table 1. The spectral map of this microquasar at a distance of 2.5 kpc is more strongly curved than of the AGNs in Fig. 1 and Fig. 2, indicating that the curvature of the AGN spectra is intrinsic as well, generated by the Boltzmann factor of the thermal electron populations, cf. caption to Fig. 2.
Fig. 4. γ-Ray broadband of the TeV source HESS J1801-233 and the associated EGRET source 3EG J1800-2338 at the northeast boundary of supernova remnant W28. Data points from [35], also see [36] and [37]. Notation as in Fig. 1. The nonthermal cascade ρ_{1} admits a power-law slope ∝E^{1−α}, α≈1.6, adjacent to the MeV–GeV plateau typical for tachyonic cascade spectra [11] and [12]. A spectral break at m_{t}γ_{1}≈5.8 GeV is visible as edge in the longitudinal component. The curvature of the thermal cascade ρ_{2} in the MeV range is due to the exponential cutoff at (m_{t}/m)kT≈20 MeV, cf. Table 1.
The spectral average of the radiation densities (3) is carried
out as
The average (6) can be
reduced to the fermionic spectral functions
which separates the spectrum into a low- and high-frequency band [30]. By making use of the spectral functions (8), we can write the averaged radiation densities (6) as
with in (7) and ω_{1} in (9), so that . The superscripts T and L denote the transversal and longitudinal radiation components, cf. (3). The spectral functions F^{T,L}(ω,γ_{1}) in (10) are obtained by substituting radiation densities (3) into the integral representation (8),
where the weight factors f_{k} read
with density dρ_{F}(γ) in (5).
The quasiclassical fugacity expansion of the spectral
functions (11) is found by
expanding density (5) in ascending
powers of .
In leading order, dρ_{F}(γ)dρ_{α,β}(γ),
The classical limit of the fermionic spectral functions F^{T,L}(ω,γ_{1})
in (8) is the
Boltzmann average [31]
where is the classical limit of the break frequency (9).
3. Spectral curvature of Galactic and extragalactic TeV γ-ray sources: Does distance matter?
The spectral fits of the active galactic nuclei (AGNs) and the
Galactic TeV sources in Fig. 1, Fig. 2, Fig. 3 and Fig. 4 are based
on the E^{2}-rescaled
flux densities
which is independent of the distance estimate in (16). Here, implies the tachyon mass in keV units, that is, we put m_{t}≈2.15 in the spectral densities (3). At γ-ray energies, only a tiny α_{q}/α_{e}-fraction (the ratio of tachyonic and electric fine structure constants) of the tachyon flux is absorbed by the detector, which requires a rescaling of the electron count n_{1}, so that the actual number of radiating electrons is , cf. Ref. [12]. We thus find the electron count as , where defines the tachyonic flux amplitude extracted from the spectral fit. (In Table 1, the subscript 1 of and has been dropped.) Electron temperature and cutoff parameter in the Boltzmann factor of density (13) are related by kT[TeV]≈5.11×10^{−7}/β, and the energy estimates of the thermal cascades in Table 1 are based on , cf. Ref. [18]. (The renormalized count is to be identified with the particle number N in the thermodynamic functions discussed in this reference.) The distance estimates of the AGNs are based on dcz/H_{0}, with the Hubble distance c/H_{0}≈4.4×10^{3} Mpc (that is, h_{0}≈0.68). Hence, d[Mpc]≈4.4×10^{3}z, and , cf. Table 1.
Electronic source distributions ρ_{i} generating the tachyonic cascade spectra of the active galactic nuclei in Fig. 1 and Fig. 2, the microquasar LS 5039 in Fig. 3, and the supernova remnant W28 in Fig. 4. Each ρ_{i} stands for a thermal Maxwell–Boltzmann density dρ_{α=−2,β}(γ) with γ_{1}=1, apart from the ρ_{1} distribution of SNR W28, which is a power-law density with α≈1.6 and γ_{1}≈2.7×10^{6}, cf. (13) and after (5). β is the cutoff parameter in the Boltzmann factor. determines the amplitude of the tachyon flux generated by the electron density ρ_{i}, from which the electron count n^{e} is inferred at the indicated distance d, cf. after (17). kT is the temperature and U the internal energy of the electron populations, cf. after (16). The parameters β and are extracted from the least-squares fit T+L in Fig. 1, Fig. 2, Fig. 3 and Fig. 4
β | d | n^{e} | kT (TeV) | U (erg) | ||
---|---|---|---|---|---|---|
1ES 0229+200 | z≈0.140 | |||||
ρ_{1} | 5.97×10^{−10} | 6.6×10^{−5} | 620 Mpc | 1.4×10^{57} | 860 | 5.8×10^{60} |
ρ_{2} | 1.79×10^{−8} | 7.5×10^{−4} | 1.6×10^{58} | 29 | 2.2×10^{60} | |
1ES 0347-121 | z≈0.188 | |||||
ρ_{1} | 5.38×10^{−10} | 2.6×10^{−5} | 830 Mpc | 1.0×10^{57} | 950 | 4.6×10^{60} |
ρ_{2} | 1.13×10^{−8} | 2.3×10^{−4} | 8.9×10^{57} | 45 | 1.9×10^{60} | |
LS 5039 (sup. conj.) | 2.5 kpc | |||||
ρ_{1} | 3.36×10^{−10} | 6.4×10^{−5} | 2.3×10^{46} | 1500 | 1.7×10^{50} | |
ρ_{2} | 1.13×10^{−8} | 2.2×10^{−4} | 7.9×10^{46} | 45 | 1.7×10^{49} | |
SNR W28 | 1.9 kpc | |||||
ρ_{1} | – | 2.2×10^{−3} | 4.6×10^{47} | – | – | |
ρ_{2} | 1.08×10^{−4} | 3.1×10^{−2} | 6.4×10^{48} | 4.7×10^{−3} | 1.5×10^{47} |
Fig. 1 shows the tachyonic spectral map of the BL Lacertae object (BL Lac) 1ES 0229+200, located at a redshift of z≈0.140, cf. Refs. [1], [2], [3] and [4]. The flux points were obtained with the HESS array of atmospheric Cherenkov telescopes in the Khomas Highland of Namibia [4]. The χ^{2}-fit is done with the unpolarized tachyon flux T+L, and subsequently split into transversal and longitudinal components. The differential flux is rescaled with E^{2} for better visibility of the spectral curvature. Temperature and source count of the electron populations generating the cascades are recorded in Table 1. In Fig. 2, we show the spectral map of the BL Lac 1ES 0347-121, at a redshift of z≈0.188, cf. Refs. [5], [6] and [7]. TeV γ-ray spectra of BL Lacs are usually assumed to be generated by inverse Compton scattering or proton–proton scattering followed by pion decay [8]. Both mechanisms result in a flux of TeV photons, assumed to be partially absorbed by interaction with infrared background photons owing to pair creation, so that the intrinsic spectrum has to be reconstructed on the basis of intergalactic absorption models depending on vaguely known cosmological input parameters [9]. The extragalactic tachyon flux is not attenuated by interaction with the background light, there is no absorption of tachyonic γ-rays. The superluminal Proca field (1) is minimally coupled to the electron current, and does not directly interact with electromagnetic radiation.
The spectral curvature apparent in double-logarithmic plots of the E^{2}-rescaled flux densities (16) is intrinsic, caused by the Boltzmann factor of the electron populations generating the tachyon flux. The curvature present in TeV γ-ray spectra does not increase with distance, at least there is no evidence to that effect if we compare the spectral slopes of the blazars in Fig. 1 and Fig. 2, and the spectral maps of other flaring AGNs such as H1426+428 at z≈0.129 and 1ES 1959+650 at z≈0.047, cf. Ref. [11]. Further evidence for intrinsic spectral curvature is provided by Fig. 3, depicting the spectral map of the microquasar LS 5039, a compact object orbiting a massive O-star [32], [33] and [34]. The spectral curvature of this Galactic binary is even more pronounced than of the BL Lacs in Fig. 1 and Fig. 2. The same holds true for the HESS spectral map of LS 5039 at the inferior conjunction studied in Ref. [18].
Fig. 4 depicts the γ-ray wideband of the TeV source HESS J1801-233 and the coincident EGRET point source 3EG J1800-2338 [35], [36] and [37]. The extended TeV source is located on the northeastern rim of the supernova remnant (SNR) W28, a mixed-morphology SNR interacting with molecular clouds [38] and [39]. The spectral map of SNR W28 in Fig. 4 is to be compared to the unpulsed γ-ray spectrum of the Crab Nebula, cf. Fig. 1 in Ref. [11], the spectral map of SNR RX J1713.7-3946 in Fig. 2 of Ref. [11], and in particular to the unidentified TeV source TeV J2032+4130 in conjunction with the associated EGRET source 3EG J2033+4118, cf. Fig. 6 of Ref. [12]. A spectral plateau in the MeV to GeV range occurs frequently in spectral maps of TeV γ-ray sources, and can easily be fitted with tachyonic cascade spectra, in contrast to electromagnetic inverse-Compton fits. SNR W28 is located at a distance of 1.9 kpc [38]. The thermal cascade in the MeV range (ρ_{2} in Fig. 4) preceding the spectral plateau is strongly curved; the power-law slope of the ρ_{1} cascade also terminates in exponential decay, but outside the presently accessible TeV range shown in the figure. The cutoff temperature of the nonthermal electron density generating cascade ρ_{1} is too high to bend the power-law slope in the TeV range covered in Fig. 4, so that the internal energy of the shock-heated plasma could not be determined from the χ^{2}-fit, in contrast with the power-law index α and the threshold Lorentz factor γ_{1}, cf. caption to Table 1.
4. Conclusion
The spectral maps of two TeV γ-ray blazars have been fitted with tachyonic cascade spectra and compared to a Galactic γ-ray binary and supernova remnant. Table 1 contains estimates of the thermodynamic parameters of the electron populations generating the superluminal cascades. The spectral curvature is intrinsic and reproduced by the tachyonic spectral densities (3) averaged with ultra-relativistic thermal electron distributions, cf. Fig. 1, Fig. 2 and Fig. 3. The shocked electron plasma in SNR W28 requires a nonthermal power-law distribution to adequately reproduce the TeV cascade in Fig. 4. The curvature in the γ-ray spectra of BL Lacs is uncorrelated with distance, so that absorption of electromagnetic radiation due to interaction with infrared photons is not an attractive explanation of spectral curvature. By contrast, there is no attenuation of the extragalactic tachyon flux, as tachyons cannot interact with cosmic background photons, so that the observed cascades are the intrinsic spectrum. The distance of the AGNs does not show in the spectral curvature, as tachyonic cascades are unaffected by background photons.
Acknowledgements
The author acknowledges the support of the Japan Society for the Promotion of Science. The hospitality and stimulating atmosphere of the Centre for Nonlinear Dynamics, Bharathidasan University, Trichy, and the Institute of Mathematical Sciences, Chennai, are likewise gratefully acknowledged.